BABYLONIA AND EGYPT 31 



Again : " Rule for dividing 700 loaves among four persons, f for 

 one, | for the second, J for the third, J for the fourth, . . . Add 

 f , |, |, and | that gives 1 + i + i Divide 1 by 1 + i + J that 

 gives i + A. Make J + ^ of 700 that is 400." Thus, to 

 modernize this solution, the four persons A, B, C, and D receive 

 on one round 1 + i + i = T loaves ; the number of rounds is 



- j r or X 700 = 400, from which the respective 



1+2+4 1+2+4 



shares are readily obtained. 



Certain problems show an acquaintance with arithmetic and 

 geometric progressions. Thus, for example, a series is given of 

 the numbers 7, 49, 343, 2401, 16807, the successive powers of 7, 

 accompanied by the words person, cat, mouse, barley, measure. 

 Almost 4000 years later this was interpreted to mean : 7 persons 

 have each 7 cats, each cat catches 7 mice, each mouse eats 7 stalks 

 of barley, each stalk can yield 7 measures of grain ; what are the 

 numbers and what is their sum ? 



Special symbols are used for addition, subtraction, and equality. 

 The Egyptian seems never to have had a multiplication table. 

 Multiplication by 13, for example, was accomplished by repeated 

 doubling, and then by adding to the number itself, its products by 

 4 and by 8. 



Herodotus reports from the fifth century B.C. that the Egyptians 

 reckoned with stones, a practise independently developed in many 

 lands, notably in the form of the abacus. This little comput- 

 ing machine of beads on wires was invented independently in 

 different parts of the ancient world. In China and other parts 

 of the Orient it is still widely and very skilfully employed. 



The handbook of Ahmes is also rich on the geometrical side. 

 It contains information in regard to weights and measures, and 

 treats of the conversion from one denomination into another. 

 As in case of the progressions, geometrical proHems are given, de- 

 pending on the use of formulas not derived in thatext itself. They 

 include computation of areas of fields bounded either by straight 

 lines or circular arcs, including in the former case owy isosceles tri- 

 angles, rectangles, and trapezoids. An isosceles trgle of base 4 



